If I understand you correctly: the Earth is way way way way way way way way to big for something like an orbital tether too have any real effect on its movement. Besides the fact that due to the counterweight there's minimal tension on the ground.

I almost wrote "nothing man made", which is true for anything we can do now. But if you think scifi enough there's little that can't be done. Turning the Earth inside out is quite possible for certain levels of civilization.

I'm not quite sure I am parsing your question correctly, but:

The Moon is yanking the Earth around, but even something that stupendously massive doesn't do much damage (the tidal forces are nothing to sneeze at.. but the 'wobble' caused by the difference between the Earths center of mass and the Earth-Moon barycenter (the point that the Earth and Moon both orbit) is negligible in nearly every consideration).

This would absolutely not be a concern.

Actually if you look at the orbital mechanics, it would offset the center of mass for the earth in relative proportion to its mass. That said, the Earth has a lot of mass and so that effect would certainly be imperceptible and probably unmeasurable (although things I used to think we'd never measure directly we seem to do so now). It would do any "damage" in the sense you are using the concept.